Statistical Quality Control (SQC) : Control chart; Major parts, Theoritical basis of 3 σ Control Limits, Types of Control chart

 

Control Charts: ©

  The control charts are the graphic devices developed by Shewhart for detecting the unnatural pattern of variations in data resulting from the repetitive process. Shewhart's control chart provides a powerful tool to detect the presence or absence of assignable causes & it enables us to control the process at desired level.

In industry we have to face two kinds of problems.

i) Whether the process confirms to given standard &

ii) To improve the level of standard by reducing variability of quality.

Both the problems can be answered by using Shewhart's control charts.

 

Major Parts of Control Charts:

 

i) Quality scale:

 It is a vertical scale marked according to the quality characteristics of each sample.

ii) Sample numbers:

The samples to be plotted on the control charts are numbered individually &   consecutively on the horizontal line, which is stressed at the bottom of the control chart.

iii) Horizontal lines (Control lines): There are three horizontal lines.

a) Central line (C.L.) - This indicates the desired standard or level of the process.

b) Upper Control line (U.C.L.) - This is the line above central line, which is usually    3-σ distance above central line. It indicates the upper limit of tolerance. 

c) Lower Control line (L.C.L.) -This is the line below central line, which is usually     3-σ distance below central line. It indicates lower limit of tolerance.

d) Plotted samples- Instead of plotting, individual items of the sample on the control chart, only the quality of entire sample represented by a single value is plotted.

 A typical control chart looks like as the figure below.

 


A point outside the control limits shows the lack of control situation.

 3-σ Control limits (Theoretical basis):

 3-σ Control limits were proposed by Dr. Shewhart for the control chart.

 Let T = t(x1, x2,…,xn) is a function of sample observations and let

                    E (T) = µt   and   V (T) = σ2t .

If T is normally distributed, then P (| (T- µt )/σt | ≤ 3) =0.9973 (from normal table).

Thus the probability that t will fall outside (µt - 3σt, µt + 3σt) is 0.0027 which is very small. In other words if variable quality characteristic is normally distributed and no assignable causes are present in the process only 27 out of 10000 values of quality characteristics will fall outside the 3-σ limits.  Since 27 out of 10000 is very small and hence negligible. In the absence of assignable causes, it is expected that all the values of quality characteristics should lie within 3-σ limits. If at least one value lies outside the  3-σ limits one suspect the presence of assignable causes and says that the process is out of control.

If the statistic does not follow normal distribution, then by Chebyshev's inequality for any K > 0,

P(|(T- µt ) | ≤ Kσt) ≥ 1 – 1/ K2

By taking K = 3,

P(|(T- µt )/| ≤ 3σt) ≥  8/9  which is fairly high probability. Thus, even if the quality characteristic does not follow normal distribution, the 3-σ control limits can be used.

 


Thus, the central line is at µt, that is        CL = µt

                                UCL = µt + 3σt      ---- (1)

                                LCL = µt - 3σt

Type of Control Charts

 There are two types of control charts.

1) Control charts for variables

2) Control charts for attributes

 Control charts for variables:

       Many quality characteristic of a product are measurable & can be expressed in specific units of measurement, such as diameter of a screw, length of rod, life of an electric bulb, specific resistance of a wire. Thus they are of continuous type & are regarded to follow normal probability law. For quality control of such data, mean &  R charts are used.

 Control charts for attributes:

        In some cases the quality characteristic cannot be measured but can be identified whether the characteristic is person or absent in the item i.e. , we can classify the product as defective or non defective Such characteristics are called as attributes We may calculate the number of defective units (d) in a sample or the number of defects in a unit. We can determine the proportion of items possessing the attribute and we use the control chat for traction defective or 'p' chart and number of defectives control chart (np-chart). Here d follows binomial distribution. If we count the number of defects in a unit, then number of defects per unit or C- chart is used Here, C follows Poison distribution.©






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